Pedro Salomão, IME/USP
Global properties of closed geodesics on the 2-sphere
If a smooth Riemannian metric on $2$-sphere has curvature sufficiently close to 1 then all closed geodesics are either (i) short and simple or (ii) long and with many self-intersections (Ballmann 1983). We extend this type of result for $r$-reversible Finsler metrics with flag curvature in the interval $(A(r),1]$, showing the non-existence of closed geodesics with one self-intersection. Here, $A(r)$ depends on the reversibility $r$ of the metric. This result is sharp in the following way: for any $e \gt 0$, we present $r$-reversible Finsler metrics with curvature in $[A(r)-e,1]$ with closed geodesics containing exactly one self-intersection. The main tools used come from the theory of pseudo-holomorphic curves in symplectizations of contact manifolds. This is a joint work with U. Hryniewicz (UFRJ).